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A245372
Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing four primes only.
1
46, 76, 96, 106, 134, 142, 146, 204, 218, 276, 310, 408, 438, 466, 518, 534, 536, 546, 580, 624, 650, 672, 680, 694, 792, 800, 896, 970, 1000, 1016, 1100, 1160, 1170, 1318, 1344, 1358, 1364, 1384, 1470, 1480
OFFSET
1,1
COMMENTS
Prime-partitionable numbers are defined in A059756.
To demonstrate that a number is prime-partitionable a suitable 2-partition {P1, P2} of the set of primes < a(n) must be found. In this sequence we are interested in prime-partitionable numbers such that the smallest P1 contains 4 odd primes.
Conjecture:
If P1 = {p1a, p1b, p1c, p1d} with p1a, p1b, p1c and p1d odd primes and p1a < p1b < p1c < p1d then the union of the integer solutions to the ten equation groups below, {{m1}, {m2}, {m3}, {m4}, {m5}, {m6}, {m7}, {m8}, {m9}, {m10}}, contains all even members of {a(n)}:
m1 = v1*p1a+1 = v2*p1b+p1a = v3*p1c+p1b = v4*p1d+p1c
m2 = v5*p1a+1 = v6*p1b+p1a^2 = v7*p1c+p1b = v8*p1d+p1a
m3 = v9*p1a+1 = v10*p1b+p1a^3 = v11*p1c+p1a^2 = v12*p1d+p1a
m4 = v13*p1a+1 = v14*p1b+p1c = v15*p1c+p1a = v16*p1d+p1b
m5 = v17*p1a+1 = v18*p1b+p1c = v19*p1c+p1a^2 = v20*p1d+p1a
m6 = v21*p1a+p1b = v22*p1b+1 = v23*p1c+p1a = v24*p1d+p1c
m7 = v25*p1a+p1b = v26*p1b+1 = v27*p1c+p1a^2 = v28*p1d+p1a
m8 = v29*p1a+p1b = v30*p1b+p1c = v31*p1c+1 = v32*p1d+p1a
m9 = v33*p1a+p1c = v34*p1b+1 = v35*p1c+p1b = v36*p1d+p1a
m10 = v37*p1a+p1c = v38*p1b+p1a = v39*p1c+1 = v40*p1d+p1b
where the vi, i = 1..40 are constrained odd naturals.
LINKS
W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
R. J. Mathar and M. F. Hasler, Is 52 prime-partitionable?, Seqfan thread (Jun 29 2014), arXiv:1510.07997
W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.
EXAMPLE
a(1) = 46 because A245602(5) = 46 and the 2-partition {3, 19, 37, 43} {2, 5, 7, 11, 13, 17, 23, 29, 31, 41} of the set of primes < 46 demonstrates it.
PROG
(PARI)
prime_part(n)=
{
my (P = primes(primepi(n-1)));
for (k1 = 2, #P - 1,
for (k2 = 1, k1 - 1,
for (k3 = 1, k2 - 1,
for (k4 = 1, k3 - 1,
mask = 2^k1 + 2^k2 + 2^k3 + 2^k4;
P1 = vecextract(P, mask);
P2 = setminus(P, P1);
for (n1 = 1, n - 1,
bittest(n - n1, 0) || next;
setintersect(P1, factor(n1)[, 1]~) && next;
setintersect(P2, factor(n-n1)[, 1]~) && next;
next(2)
);
print1(n, ", ");
);
);
);
);
}
# PP = {{2x, x = 1:1000} - {A245664(n), n = 1:145}
# - {A249302(n), n = 1:77}}
PP = [2, 4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, \
38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, \
...
1994, 1996, 1998, 2000];
for(m=1, #PP, prime_part(PP[m]));
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved